JEE Main & Advanced Mathematics Pair of Straight Lines Question Bank Bisectors of the angle between the lines, Point of intersection of the lines

  • question_answer If the equation \[a{{x}^{2}}+2hxy+b{{y}^{2}}=0\]has the one line as the bisector of angle between the coordinate axes, then [Bihar CEE 1990]

    A)            \[{{(a-b)}^{2}}={{h}^{2}}\]     

    B)            \[{{(a+b)}^{2}}={{h}^{2}}\]

    C)            \[{{(a-b)}^{2}}=4{{h}^{2}}\]   

    D)            \[{{(a+b)}^{2}}=4{{h}^{2}}\]

    Correct Answer: D

    Solution :

               Let one line be \[x+y=0\,\,\,\Rightarrow \,\,{{m}_{1}}=-1\]            and we know that \[{{m}_{1}}+{{m}_{2}}=-\frac{2h}{b}\]                   .....(i)            and \[{{m}_{1}}{{m}_{2}}=\frac{a}{b}\]                                                       .....(ii)            Therefore from (ii), \[{{m}_{2}}=-\frac{a}{b}\]            \[\Rightarrow -1-\frac{a}{b}=\frac{-2h}{b}\Rightarrow {{(a+b)}^{2}}=4{{h}^{2}}\].

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